Interpolation Models with Multiple Hyperparameters

Abstract

A traditional interpolation model is characterized by the choice of regularizer applied to the interpolant, and the choice of noise model. Typically, the regularizer has a single regularization constant α, and the noise model has a single parameter β. The ratio α/β alone is responsible for determining globally all these attributes of the interpolant: its ‘complexity’, ‘flexibility’, ‘smoothness’, ‘characteristic scale length’, and ‘characteristic amplitude’. We suggest that interpolation models should be able to capture more than just one flavour of simplicity and complexity. We describe Bayesian models in which the interpolant has a smoothness that varies spatially. We emphasize the importance, in practical implementation, of the concept of ‘conditional convexity’ when designing models with many hyperparameters.